Mathematics to the Rescue of Democracy: What does Voting Mean and How can it be Improved?

By Paolo Serafini

This book explains, in a straightforward way, the foundations upon which electoral techniques are based in order to shed new light on what we actually do when we vote. The intention is to highlight the fact that no matter how an electoral system has been designed, and regardless of the intentions of those who devised the system, there will be goals that are impossible to achieve but also opportunities for improving the situation in an informed way. While detailed descriptions of electoral systems are not provided, many references are made to current or past situations, both as examples and to underline particular problems and shortcomings. In addition, a new voting method that avoids the many paradoxes of voting theory is described in detail. While some knowledge of mathematics is required in order to gain the most from the book, every effort has been made to ensure that the subject matter is easily accessible for non-mathematicians, too. In short, this is a book for anyone who wants to understand the meaning of voting.  

• Language: English
• Publisher: Springer
• Release date: Mar 2, 2020
• ISBN: 9783030383688

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© Springer Nature Switzerland AG 2020

P. SerafiniMathematics to the Rescue of Democracyhttps://doi.org/10.1007/978-3-030-38368-8_1

1. Introduction

Paolo Serafini¹  

(1)

Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy

Paolo Serafini

Email: [e-mail]

In 1925, the famous and influential journalist Walter Lippmann  wrote in his book The Phantom Public ([59] p. 46): What in fact is an election? We call it an expression of the popular will. But is it? We go into a polling booth and mark a cross on a piece of paper for one of two, or perhaps three or four names. Have we expressed our thoughts on the public policy of the United States? Presumably we have a number of thoughts on this and that with many buts and ifs and ors. Surely the cross on a piece of paper does not express them. It would take us hours to express our thoughts, and calling a vote the expression of our mind is an empty fiction.

Lippmann was right. The present way of voting is rough and may produce electoral outcomes that are puzzling either because they don’t seem to reflect the wishes of the majority of voters or because they lead to deadlock situations.

Voting is a fundamental activity in the life of a society, and it would be appropriate to know exactly what kind of outcome we want from an election, what the theoretical limitations of voting are, what kind of information would be helpful for the electors to pass along and, finally, how the efficacy of voting could be improved.

In general, people have some minimum knowledge about medicine, economics or other disciplines, and this enables them, at least partially, to direct their actions. Instead, in the practice of voting, the impression is that there is very little to understand, because everything seems obvious and elementary. On the contrary, it is much more complex than it appears.

The aim of this short book is to explain, in a hopefully simple way, the foundations upon which electoral techniques are based. Although one may think that voting pertains social or juridical sciences, the fact is that the basis of every electoral system is mathematics. What’s more, it is not a particularly abstruse form of mathematics. It can be made comprehensible to non-mathematicians. But neither is it elementary. In any case, the effort to understand it can garner the significant reward of a greater awareness of what we do when we vote. The title of the book highlights this hope.

The aim is not to provide a detailed description of certain electoral systems, either in Italy or elsewhere. There will, however, be more than a few references to existing or past situations, either as examples or to underline problems and defects.

It is important to know that, no matter how an electoral system has been devised, there are goals that are impossible to pursue, independent of the presumptions of the system’s authors, but there are also opportunities toward improving the situation in an informed way.

It may be surprising to see that many electoral systems that are currently employed in various countries have evident defects and, yet are, nonetheless, still in use, justified for reasons of habit and familiarity with a particular method. For instance, one of the most fallacious ideas is that the alternative that receives most votes should be chosen even in cases in which it has received less than half of the total votes. Yet, many electoral systems are based on this idea.

A quite common bit of confusion is related to the aim of voting. Voting for parliamentary representation is very different from voting for a president or a govern. If these two aims overlap, the outcomes can be problematic.

The vote that we are accustomed to is the simple indication of a preference for one alternative among many. What we think of the alternatives that we have not chosen is not transmitted as information, and this is unfortunate, because the vote outcome would be in far greater compliance with the collective will than the current method of voting. Is it conceivable to modify a secular habit? However difficult this notion may seem, the time has probably come to make the vote more informative. In this book, we will try to illustrate the advantages of voting in a more completely engaged way. To this purpose, we will make explicit reference in Chap. 8 to the method devised by Balinski and Laraki [12, 15], the so-called Majority judgement.

The argument is a very large one and, in order to be thorough, four times as many pages would be necessary. Hence, we have opted to focus on a few select principles in order to make the fundamental aspects understandable and possibly direct the reader to more specific essays for an in-depth study.

© Springer Nature Switzerland AG 2020

P. SerafiniMathematics to the Rescue of Democracyhttps://doi.org/10.1007/978-3-030-38368-8_2

2. Voting

Paolo Serafini¹  

(1)

Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy

Paolo Serafini

Email: [e-mail]

In the idea of the State that we inherited from the Age of Enlightenment, there are three public powers that belong to the citizens who constitute the State: the legislative power, the executive power and the judicial power. Clearly, there are other powers in the society that can condition, for better or for worse, the life of the State, but it is the three aforementioned ones that belong to the community of citizens. The practice of these powers cannot be carried out by the community as a whole, and thus it is necessary to delegate these tasks.¹

If we want the three powers to be independent of each other, it is important to understand how the machinery of mandate allows for such independence. There can be differences among nations. For instance, the Italian Constitution guarantees the independence of the judicial power from the other two in a precise way. The mandate goes through a selection that is an expression of the citizens inasmuch it is guaranteed by public laws. It is not the expression of a direct vote. If it were so, it would follow the same logic of the mandate for the legislative and executive powers, thereby jeopardizing the very concept of independence. This separation of the judicial power from the other two is not as sharp in every nation and the very fact that there are often attempts to weaken the separation shows why it is so necessary. 

The question is different for the legislative and executive powers. The mandate is direct and is carried out through a vote. The choice of how to implement the mandate changes according to the nation. Basically, there are three possibilities: the vote is for the legislative power and the mandate for the executive power is left to the legislative power, or the vote is for the executive power and the mandate for the legislative power is left to the executive power, or there are two distinct votes, one for the legislative power and another for the executive power.

The first method is the one used in so-called parliamentary democracies: a parliament is elected and the executive power is the expression of the will of the parliament. The second method is not formally present in any democratic country. However, it can, in practical terms, happen. The recent history of the Italian parliament shows that an inversion of the two roles has been attempted. The third method is in effect in a number of important countries, like France and the United States of America.

Having an ‘effective’ mandate for both the legislative power and the executive power at the same time is problematic, because the legislative power is well delegated when the society’s many articulated requests are being addressed, whereas the executive power is well delegated if the government is granted a certain degree of stability, at least for a set period of time. It is not possible to govern well if the government’s actions are frequently interrupted or weakened. 

Representation and governmental stability are therefore the two opposing requirements of a parliamentary democracy. If the method of voting favors representation, the governing becomes less stable and, vice versa, if governmental stability is favored, the very essence of the legislative power is undermined. In designing a voting method, it must be clear which goal we want to pursue, within the bounds of the constitutional law.

The Italian Constitution has been designed to give ample representation to the citizens, after two decades of dictatorship, with the idea in mind that the parliament would be able to form stable governments. Indeed, this is what happened from 1948 (when the Constitution was approved) until the beginning of the nineties. A fact that has often been misunderstood concerns the large numbers of different governments that have been formed throughout these years. In most cases, each change of government consisted in a turnover of power positions in the administration. The political course of action, contrastingly, was always stable. Things began to change starting with the 1994 elections, when the electoral mechanism was forced to address the choice of the prime minister as well, in spite of the fact that the Constitution had not been changed. This created enormous confusion that has still not dispelled. The idea has been spread that political elections are meant for choosing governments directly. It has been presumed that the vote percentage ranking could constitute the basis for obtaining a government mandate,² and there has been a lot of misguided talk about ‘winners’ and ‘losers’, terms that do not exist in the Constitution.

The mechanisms for forming a government in a parliamentarian democracy are not based on rankings. The scenario involves parties that have been apportioned a certain number of seats and the goal is to form coalitions that can gather at least half of the total number of seats (unless a single party gets the absolute majority of seats, a very unlikely possibility for many countries, let alone Italy). How this can happen is more pertinent to Game Theory rather than it is to Social Choice Theory. For this purpose, particular indices, like the Shapley–Shubik value [91, 92], have been proposed. These indices evaluate the strength of the parties toward the formation of a coalition. It should not be surprising that three parties that have respective seat strengths of 45, 35 and 20 $$\%$$ actually have the same bargaining power, because they are all equally necessary to obtain the absolute majority of seats.

All of these aspects should be clear in the choice of an electoral mechanism and in the consequent expectation within the population. If the need to entrust the executive power to someone by a direct election is felt to be mandatory, then it seems unavoidable that two separate elections will be necessary, one for the parliament and the other one for the government (in Italy, this would entail a modification of the Constitution). The United States of America have been governed in this way for more than two centuries.

By having separate voting for the legislative and executive powers, there is the risk that the political colors of the two powers will be different, thus creating a disharmony between government and parliament. When this has, indeed, happened, like, for instance, in France with the so-called ‘cohabitation’, or in the United States with a President at odds with a hostile Congress, no insurmountable problems have occurred. Rather, one gets the impression of a healthy democratic dialectic. It seems more problematic to have a unique vote and compel the parliament to form difficult coalition agreements. The recent examples of Germany, Italy, Spain and Israel demonstrate this fact.

In the subsequent chapters, we shall examine the problems that arise with the choice mechanism, by dealing separately with the questions concerning the formation of a government by electoral methods (Chaps. 3, 4, 5, 6, 7 and 8) and those pertinent to having parliamentary representation (Chaps. 9, 10, 11, 12, 13). We will not deal with the issues connected to the formation of coalitions in parliaments. In Chap. 14, we examine a common feature of both aspects, namely, measuring the voting power of a single voter.

As for the first issue, that is, the choice of a government, the problem of ranking all present alternatives is strictly related. This wider issue is not very relevant in a political context, but it is important in sport, in which ranking is essential. As we shall see in the subsequent chapters, from a mathematical point of view, declaring the winner of a political contest is the same as declaring the winner of some sport competitions.

An issue that must be considered in the future concerns the type of information that electors express by voting. Presently, they express only a preference for one particular alternative, with nothing been said about all of other alternatives. This is clearly very simple and easy to be carried out by all citizens. However, the missing information on the alternatives may cause paradoxical consequences.

A choice method that avoids many of the paradoxes and impossibilities present nowadays in the voting systems, even in more detailed voting methods, is the already cited Majority judgement [12, 15], which might represent an extremely interesting electoral method for the future.³ In Chap. 8, we shall present the method in great detail by showing how it works and listing its limitations. This method is currently applied in small contests, like sport competitions, wine contests, in which only a few judges have to express a vote, or, put better, a judgement. Transferring this method into a political competition with millions of voting people, might be not so simple. Perhaps, it is only a problem of ‘adjustment’. Also, changing an electoral system every few years (like in Italy) creates adjusting difficulties.

The considerations that can be drawn for an electoral system nationwide also hold, on a smaller scale, for choices made at local level. However, even if the same pattern of executive and legislative power is replicated at the local level (the judicial power, at least in Italy, is not a competence of local authorities), one should note that the legislative scope of a local assembly is necessarily quite reduced, whereas the executive scope is not. Therefore, in order to select a local electoral system, the considerations related to a choice of executive type become predominant.

Very often, proposals are brought forth in Italy to write an electoral law, similar to the one for mayoral elections mandating that the name of the future Prime Minister be known as soon as the polls are closed.⁴ This would mean completely forgetting about the need for legislative representation at the national level. The national level is different from the local level.

Footnotes

1

Concerning the idea of ‘direct democracy, without mandate and intermediate bodies, see the considerations in the last Chap. 15.

2

On the danger of considering relative majorities as being eligible to form governments, see Chap. 6.

3

It is perhaps useful to read the comments on   Majority judgement by three Nobel prize winners in Economics, as quoted in [1]: K.J. Arrow (Nobel prize 1972): ‘The authors have proposed a very interesting voting method to remedy the well-known defects in standard methods, such as plurality voting. It requires the voters to express their preferences in a simple and easily comprehensible way, and the authors supply evidence that the candidate chosen by their methods is a reasonable selection. This work may well lead to a useful transformation in election practice’. R. Aumann (Nobel prize 2005): ‘Michel Balinski has done it again! He has produced—this time with Rida Laraki—a beautiful, comprehensive, conceptually deep, and utterly sound treatise on the mechanics of democracy. By no means an abstruse, ivory-tower exercise in pure math, the work is supported by a plethora of in-depth empirical analyses taken from real life. Most important, the book introduces a vital new idea that promises to revolutionize democratic decision making: ‘judging’ rather than voting. Enjoy—while learning’. E.S. Marskin (Nobel prize 2007): ‘Balinski and Laraki propose an intriguing new voting method for political elections: have voters ‘grade’ the candidates and elect the one having the highest median grade. The method will be controversial but deserves careful consideration.’

4

Mario Monti has dubbed this need as  ‘Sunday night fever’.

© Springer Nature Switzerland AG 2020

P. SerafiniMathematics to the Rescue of Democracyhttps://doi.org/10.1007/978-3-030-38368-8_3

3. Aggregating Different Evaluations into One Unique Evaluation

Paolo Serafini¹  

(1)

Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy

Paolo Serafini

Email: [e-mail]

When we deal with the choice of a president or a government through an election, the problem that we have to face is fundamentally that of determining a unique collective evaluation that represents, in the best possible way, an aggregation of the individual evaluations expressed by the voters. We can see that this problem is by no means easy. Moreover, aggregating methods that seem appealing and easily understandable may be subject to serious malfunctions, as we shall see.

If we want to formalize the aggregation problem, we have to introduce ‘candidates’ (or ‘alternatives’, as we shall also refer to them) and ‘electors’ (or ‘voters’ or ‘judges’). Electors express their individual evaluations on the candidates. Starting from all individual evaluations, we have to build up a mechanism that produces a unique collective evaluation on all of the candidates.

We have to specify what type of collective evaluation we want to produce and what type of individual evaluation we want to input into the mechanism. According to the goal that we are pursuing, we have available different alternative ways to define both the collective evaluation and the individual evaluation. The mechanism that aggregates the individual evaluations into the collective evaluation clearly depends